Questions involving philosophy of mathematics

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5
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1answer
54 views

Presentation of a group question

So I know that given a presentation of a group $G$, one can derive from the relations of the group presentation any element in the group $G$ right. However, I do have some confusion. If we take ...
3
votes
2answers
111 views

Naturality in linear algebra

Question. How can we formalize these intuitions about predicates on matrices? Let $P$ denote a predicate on matrices, so that $P(A)$ is true for some choices of matrix $A$ and false for all ...
1
vote
0answers
30 views

Imagening the Thurston geometries

I can (more of less) imagine how it would look if space was Euclidean, spherical of hyperbolic. But there are 8 Thurston geometries see https://en.wikipedia.org/wiki/Geometrization_conjecture how ...
3
votes
2answers
257 views

Why is it important, that mathematics can be formalized in set theory?

Why is it important, that mathematics can be formalized in set theory? As one can read in the thread Are there areas of mathematics that cannot be formalized in set theory? Today known mathematical ...
105
votes
5answers
3k views

Why can't differentiability be generalized as nicely as continuity?

The question: Can we define differentiable functions between (some class of) sets, "without $\Bbb R$"* so that it Reduces to the traditional definition when desired? Has the same use in at least ...
5
votes
1answer
115 views

What are numbers? [duplicate]

The title is a bit of clickbait, but I think it's justified. How did I came to ask this question In programming, many programming languages have concepts of a hierarchy of numerical types. Often ...
11
votes
2answers
232 views

Motive for the definition of inner product

Mathematicians pride themselves on writing proofs of propositions in an elegant way, but frequently (maybe even usually?) neglect to formally write motivations of definitions with the same elegance, ...
1
vote
2answers
50 views

Why are Euclid axioms of geometry considered 'not sound'?

The five postulates (axioms) are: "To draw a straight line from any point to any point." "To produce [extend] a finite straight line continuously in a straight line." "To describe a circle with ...
2
votes
2answers
79 views

How to generate complicated looking identities such as $\sqrt [3] {2 + \sqrt 5} - \sqrt [3] {2 - \sqrt 5}=1$ easily?

How to generate complicated looking identities, or even more complicated looking identies such as $\sqrt [3] {2 + \sqrt 5} - \sqrt [3] {2 - \sqrt 5}=1$ easily? I saw the identity to be shown. What is ...
-3
votes
1answer
120 views

Why a line is said to have infinite number of points? [duplicate]

Why a line is said to have infinite number of points? Is this so because a line is ever lasting or we can not count how many points does it have? Finite means: Having an end. Infinite means: No end! ...
2
votes
1answer
125 views

I need help understanding Frege's definition of number

I have really been trying to understand Frege's definition of a number or at least gain a strong intuition of it. However, my attempts have not been fruitful. If someone could help me it would be much ...
1
vote
2answers
43 views

When the probability model of an experiment is correct?

Suppose I wanted to tell what's the probability of event $A$: getting 2 tails in a row of 5 coin tosses. According to the classical definition of probability, the probability of this event is equal to ...
5
votes
2answers
81 views

What is mathematical definition of a fluid?

I am searching the precise and mathematical definition of a fluid for a long time but I did not find it anywhere. What I mean by precise and mathematical can be understood by the following: There is ...
0
votes
0answers
28 views

curves in Poincare half space (3 dimensional hyperbolic geometry)

Okay maybe I am going a bit ahead of my self The Poincare half plane still has many mysteries for me But still I was puzzeling about the 3 dimensional variant of it. So lets assume an hyperbolic 3 ...
2
votes
1answer
70 views

Principle of mathematical induction

In his book “Introduction to Mathematical Philosophy” Bertrand Russell seems to reach the conclusion that mathematical induction is a definition and not a principle. In essence he states that ...
2
votes
4answers
144 views

Peano's Axioms: Mathematical Philosophy

In Peano Axioms, why is it necessary to define number and successor. Does not using them imply that we know what they mean? Or could they have just as easily been any two arbitrary terms which are not ...
29
votes
4answers
2k views

Avoiding proof by induction

Proofs that proceed by induction are almost always unsatisfying to me. They do not seem to deepen understanding, I would describe something that is true by induction as being "true by a technicality". ...
5
votes
3answers
203 views

Is there an area of study regarding why certain mathematical definitions are useful?

Often in my studies I'll come across an definition, which I understand, and but don't necessarily see why the particular definition was chosen to be studied. For example, the topological axioms ...
2
votes
2answers
88 views

Error in Introduction to Mathematical Philosophy

Is this an error in the text or am I reading incorrectly. What am I missing? Introduction to Mathematical Philosophy Page 18 Definition of Number “A relation is said to be “one-one” when, if $x$ has ...
-1
votes
2answers
65 views

a question of infinity [closed]

If infinity or infinities cannot be physically proved ie actually counted, how do we know they really exist
3
votes
2answers
60 views

What are the uses of cross-theoretic identifications within mathematics?

I've been thinking about the identification of objects from different mathematical theories. For example, when you do set theoretic constructions of the natural numbers and identify, e.g., 0 with the ...
4
votes
2answers
107 views

Does randomness exist? [closed]

I've been plagued with this question for a few years now and wanted to know what others think. Does true randomness really exist? In mathematics, a random process is based on the concept of random ...
1
vote
1answer
85 views

Is a derivation a proof?

Is there a difference between "derivation" and "proof"? I imagine a derivation is a type of proof but that proofs are perhaps more general. Although then again, I suppose every proof should be ...
1
vote
1answer
38 views

Justifying the use of real numbers for measuring length

I am not sure if this is the most appropriate place to post this but here goes nothing: Assume we were trying to come up with system of numbers $S$ to model our intuition of length. We want $S$ to ...
0
votes
1answer
52 views

More details of the “Standard View og Proof” with three points are needed.

I have a Danish book about the theory of knowledge for mathematicians which I have tried my best to translate some parts into English. According to the lecturer, we can with "certain reasonability" ...
0
votes
0answers
42 views

Computability, Continuity and Constructivism

Triggered by an IMO extremely interesting question & Mathematics Stack Exchange, asked by Dal: Computability and continuous real functions And a link in one of the comments that could have ...
2
votes
3answers
180 views

Every planar graph can be embedded on a sphere - formal proof?

The proof of the following theorem: A graph can be embedded on the surface of a sphere without crossings if and only if it can be embedded in the plane without crossings. is very short- The ...
2
votes
1answer
48 views

Given ∼R and ∼ B, derive ∼ (R ∨ B)

*This question deals with the derivation system SD (Sentential Derivation), the rules of which can be seen on pages 3-4 here: http://www.shamik.net/teaching/materials/dasgupta%20SL%20definitions.pdf ...
4
votes
2answers
207 views

Why is homeomorphism understood as stretching and bending?

A function $f: X \to Y$ between two topological spaces $(X, T_X)$ and $(Y, T_Y)$ is called a homeomorphism if it has the following properties: $f$ is a bijection (one-to-one and onto) $f$ is ...
58
votes
11answers
4k views

How do people who study intensely abstract mathematics “imagine” or understand the concepts they are studying or being taught? [closed]

This question is probably to the actual people who study such mathematics, rather than any "third-party". I haven't studied any such mathematics, but I can imagine that some (probably most of it) of ...
0
votes
2answers
142 views

The word “times” for multiplication…? [closed]

The word "times" for multiplication operation which is quite touching to the concept of time (feeling time this way 0*1=0). When was introduced that term? Did any other language have the kind of term ...
0
votes
2answers
39 views

Question about induction to infinity with regard to Bolzano's philosophy

I'm a philosophy and mathematics student, and I'm writing a paper on a proof put forward by Bolzano that if we can know one thing to be true, then we can know infinite truths. Put simply, he states ...
17
votes
6answers
3k views

Why is mathematical induction a valid proof technique?

Context: I'm studying for my discrete mathematics exam and I keep running into this question that I've failed to solve. The question is as follows. Problem: The main form for normal induction over ...
0
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2answers
90 views

Why does randomness exhibit a pattern in the long run?

!!! Layman here so please avoid complex math and answers. Random (usually pseudorandom) events are usually characterized along these lines: Each outcome in a trial experiment must be i.i.d.; i.e. ...
0
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2answers
77 views

What is the difference between a counter-intuitive statement and a paradox?

In mathematics and logic, what is the difference between a counter-intuitive statement and a paradox? For example, what differs something like the Banach-Tarski theorem or Gabriel's horn from ...
5
votes
1answer
122 views

What's the difference between “unprovable” and “undecidable”?

It seems to me that there is a difference between an unprovable sentence, and an undecidable sentence, but sometimes I have the impression that some authors use the terms interchangeably. In my ...
3
votes
1answer
65 views

Sheaves in Philosophy

I once found a book on google.books. It was about the applications of sheave theory to philosophy or more general to social studies. I don't remember for sure. i just know it was not the book Sheaves ...
1
vote
1answer
67 views

What does if-then has to do with not being true?

I'm reading Chihara's: Constructibility and Mathematical Existence. It says: An even more radical view rejects the assumption that mathematics is true—at least in the straightforward way that ...
3
votes
5answers
220 views

Does taking courses in mathematics give any help for mathematical logic?

I'm undergraduate student of philosophy department and I think I'll major in mathematical logic. For studying mathematical logic, I thought studying math lectures would give help to logic. So I ...
25
votes
5answers
2k views

Can mathematics get from other sciences what it got from physics?

Throughout history, physics has been an unparalleled source of '' inspiration'' for discovering/inventing mathematical ideas, which is due to its ability to describe the physical world. But can this ...
1
vote
2answers
162 views

Is mathematics invented or discovered? [closed]

In physics for example, and in science in general, facts are "discovered" in the sense that they arise from observing nature. A particle is discovered if we can measure its existence in nature. A law ...
4
votes
1answer
139 views

Are there logical arguments against modern $\sf ZFC$ set theory?

As of asking this question, my knowledge of set theory is quite pedestrian. I've read about it in numerous nontechnical papers and even worked through three chapters of Jech - Set Theory, but in terms ...
4
votes
1answer
126 views

Is the anti-foundation axiom considered constructive?

In the area of theoretical computer science that I am interested in, constructive mathematics is of practical interest because it gives algorithms that can be implemented on a computer. However, ...
1
vote
2answers
82 views

philosophy : first axiom of geometry and variable curvature

The very first axiom of geometry can be described as: Two different points lay on one and only one line. And I was wondering are there surfaces where this axiom irrecoverably fails? and I found ...
35
votes
11answers
3k views

Why not both true and false?

Why can't some mathematical statement (or whatever is the correct term) be both true and false? For example we can prove (e.g. by induction) that $1+2+3+\cdots+n=\frac{n(n+1)}{2}$ for all positive ...
4
votes
6answers
2k views

What's behind the Banach-Tarski paradox? [closed]

The discovery of the Banach-Tarski paradox was of course a great thing in mathematics but raises the issue of the relation between mathematics and reality. Empirically there are good reasons for faith ...
17
votes
4answers
2k views

Why do we need to learn Set Theory?

I was planning to write some article for the Mathematics magazine of our college and it occurred to me that it will be a good idea to write about the impact and importance of Set Theory. I plan ...
4
votes
6answers
434 views

Question on induction technique

When one uses induction (say on $n$) to prove something, does it mean the proof holds for all finite values of $n$ or does it always hold when even $n$ takes $\pm\infty$?
7
votes
5answers
235 views

Why do we first introduce the open set definition for continuity instead of the neighborhood definition?

After (nearly) completing my course in topology, something weird just stuck out to me which I hadn't considered before. When first discussing continuity, we often use the following definition: Let ...
2
votes
1answer
43 views

Properties of the simplest object in n-dimension

In my boredom, I was thinking about why the simplest 3d object (i.e. the one with the least faces, sides, vertices) was the tetrahedron. After it made sense to me, I realized some cool stuff which was ...